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Let $k$ be a field. A folklore theorem states that dg-categories (over $k$), $A_{\infty}$-categories (over $k$) and stable ($k$-linear) $(\infty, 1)$-categories are "the same" (see for example Stab …

Let $\mathcal{C}$ and $\mathcal{D}$ be dg categories over a field $k$ of characteristic zero. Then one can form their tensor product $\mathcal{C} \otimes \mathcal{D}$: the objects of the tensor produc …

Let $(A, \{m_k\}_{k \geq 1} )$ be an $A_{\infty}$ algebra over a field $k$. Recall that this is the data of a $\mathbb{Z}$-graded $k$-vector space, along with a collection of $k$-nary operations which …

Let $A$ be a dg- or $A_{\infty}$-category (with $\mathbb{Z}$-graded Hom sets, over a field of characteristic $0$). Let $HH_*(A)$ be the Hochschild homology of $A$. Suppose that $HH_n(A)=0$ for all $n …